Cosmology for the HP48S/SX/G/GX

Overview

1°) Cyclic Universes: CYCL Subdirectory

a) Negative Radiation Pressure - Zero Matter
b) Negative Matter Density - Zero Pressure

2°) Empty Universes
3°) Tolman Universes
4°) Our Universe ?: UNIV Subdirectory

a) With Carlson Integrals
b) Numerical Integrations

5°) Carlson Elliptic Integrals RF RJ RFZ RJZ

-These programs employ Einstein's equations

2 R(t).d²R/dt² + (dR/dt)² + k.c² = ( -(8.PI.G/c² ).p + (Lambda).c² ).R²(t)
(dR/dt)² + k.c² = ( (8.PI.G/3) (rho) + (Lambda/3).c² ).R²(t)

    R(t) is the scale factor = the "radius" of the Universe , Lambda = cosmological constant , rho = density of matter and energy ; p = pressure

     k = +1 for Spherical   Universes
     k = 0   for Euclidean   ---------
     k = -1 for Hyperbolic ---------

-Knowing the cosmological redshift z of a galaxy, and the values of the current parameters,
the following routines compute one or several "distances", the recessional velocity and the age of the Universe.

              D = light-travel time distance ( in Giga-light-years ) = light-travel time ( in Giga-years )
             D₀ = comoving distance ( in Giga-light-years )
             D_L = luminosity-distance ( in Giga-light-years )
             V_R = recessional velocity ( speed of light = 1 )
              t₀ = age of the Universe ( in Giga-years )

     t₀ = (c/H₀)     §₀¹     y. [ (Omega)_lambda.y⁴ + ( 1-(Omega)_tot ).y² + (Omega)_mat.y + (Omega)_rad ] ^-1/2 dy
    D = (c/H₀) §_y(em)¹ y. [ (Omega)_lambda.y⁴ + ( 1-(Omega)_tot ).y² + (Omega)_mat.y + (Omega)_rad ] ^-1/2 dy           y(em) = y at the instant of emission
    D₀ = (c/H₀) §_y(em)¹      [ (Omega)_lambda.y⁴ + ( 1-(Omega)_tot ).y² + (Omega)_mat.y + (Omega)_rad ] ^-1/2 dy               z + 1 = R₀/R = 1/y_em

where (Omega)_tot = (Omega)_mat + (Omega)_lambda + (Omega)_rad

IMPORTANT NOTE:

-The variable 'H0' contains in fact 1/H₀ = 13.7717214286 giga-years, which corresponds to H₀ = 71 km/s/MPc
-Change this number if you want to use another Hubble "constant".

1°) Cyclic Universes: CYCL Subdirectory

a) Negative Radiation Pressure - Zero Matter

-With this program, you choose the minimum , current and maximum scale factor: R_min , R₀ , R_max ( in giga-light-years ) and the redshift z of a galaxy
-H₀ is computed with these data, not with the number stored in 'H0'
-In these universes, k = -1: they are hyperbolic.

STACK	INPUTS	OUTPUTS
Level 6	/	L
Level 5	/	q
Level 4	R_min	H₀
Level 3	R₀	P
Level 2	R_max	t₀
Level 1	z	D

where L = cosmological constant ( in Gy^-2 ) , q = deceleration parameter , H₀ = Current Hubble "constant" ( km/s/Mpc )
P = period , t₀ = Age of the universe ( since the last minimum scale factor ) , D = light-travel-time distance ( G-l-y )

Example:

   1 ENTER
   14 ENTER
314 ENTER
7 NRAD -> D = 12.5328286472
t₀ = 13.9689888753
P = 986.465095748
H₀ = 69.5942743888
q = -0.00313633521
L = -0.0000304268892563

Note:

-The formulas are simple: all may be expressed with trigonometric functions.
-Unfortunately, it's not always the same:

        b) Negative Matter Density - Zero Pressure

-The inputs are the same as the program above.
-You choose the minimum , current and maximum scale factor: R_min , R₀ , R_max ( in giga-light-years ) and the redshift z of a galaxy
-In these universes, k = -1: they are hyperbolic.

STACK	INPUTS	OUTPUTS
Level 6	/	L
Level 5	/	q
Level 4	R_min	H₀
Level 3	R₀	P
Level 2	R_max	t₀
Level 1	z	D

where L = cosmological constant ( in Gy^-2 ) q = deceleration parameter H₀ = Current Hubble "constant" ( km/s/Mpc )
P = period t₀ = Age of the universe ( since the last minimum scale factor ) D = light-travel-time distance ( G-l-y )

Example:

1 ENTER
14 ENTER
314 ENTER
7 NMAT -> D = 13.5614979392
t₀ = 15.4905798443
P = 993.86512963
H₀ = 67.2299023844
q = -0.036404800874
L = -0.0000303302969336

Notes:

-This program employs Carlson elliptic integrals.
-With an HP48G/GX, execution time = 11 seconds in example above

2°) Empty Universes

-With these universes, (Omega)_mat = (Omega)_rad = 0
-You choose (Omega)_lambda and the redshift z of a galaxy

STACK	INPUTS	OUTPUTS
Level 7	/	k
Level 6	/	R₀
Level 5	/	V_R
Level 4	/	t₀
Level 3	/	D_L
Level 2	(Omega)_lambda	D₀
Level 1	z	D

Example: (Omega)_lambda = 0.4 z = 7

0.4 ENTER
7 VAC -> D = 14.0146633022
D₀ = 34.6018530194
D_L = 487.804123114
t₀ = 16.2332249293
V_R = 2.51252925778
R₀ = 17.7792159139
k = -1

Notes:

-This routine is very fast.
-If the universe is cyclic ( it's the case if (Omega)_lambda < 0 ) , the period is returned in level 8
-For instance, with (Omega)_lambda = -0.4 , P = 68.4081910522

3°) Tolman Universes

-In Tolman universes, (Omega)_mat = 0

-So, you choose (Omega)_rad & (Omega)_lambda > 0

STACK	INPUTS	OUTPUTS
Level 4	/	k
Level 3	(Omega)_rad > 0	R₀
Level 2	(Omega)_lambda > 0	t₀
Level 1	z	D

Example: (Omega)_rad = 0.001 , (Omega)_lambda = 0.4 z = 7

0.001 ENTER
0.4 ENTER
7 TOL -> D = 13.9333170727
t₀ = 15.5435004234
R₀ = 17.7940504729
k = -1

4°) Our Universe (?) UNIV Subdirectory

a) With Carlson Elliptic Integrals

-You choose positive (Omega)_mat , (Omega)_lambda , (Omega)_rad and the galaxy redshift z

STACK	INPUTS	OUTPUTS
Level 7	/	k
Level 6	/	R₀
Level 5	/	V_R
Level 4	(Omega)_rad	t₀
Level 3	(Omega)_lambda	D_L
Level 2	(Omega)_mat	D₀
Level 1	z	D

Example1:     With    (Omega)_mat = 0.044 , (Omega)_lambda = 0.519 , (Omega)_rad = 0.000049   &   z = 7

     0.000049 ENTER
        0.519     ENTER
        0.044     ENTER
            7        Z->D D = 14.1190889700
D₀ = 34.1900167447
   D_L = 413.922726233
   t₀ = 15.5877288348
V_R = 2.48262477004
   R₀ = 20.8339616550
k = -1

Example2:     With    (Omega)_mat = 0.271 , (Omega)_lambda = 0.732 , (Omega)_rad = 0.000049 &   z = 7

( These Omega-values are close to those suggested by WMAP = NASA's Wilkinson Microwave Anisotropy Probe )

      0.000049 ENTER
        0.732     ENTER
        0.271     ENTER
           7         Z->D D = 12.8904186966
D₀ = 28.7574862107
   D_L = 229.550460568
   t₀ = 13.6678867998
V_R = 2.08815480039
   R₀ = 249.407504594
k = +1

Notes:

-With an HP48G/GX, execution time is about 18 seconds.
-If (Omega)_rad = 0 , choose Infinite = 9E499 ( otherwise, there would be an error )
-The precision may be small if z is very small.
-This program uses Carlson elliptic routines RFZ & RJZ.

        b) Numerical Intregration

ITG uses the built-in integral function.

STACK	INPUTS	OUTPUTS
Level 7	/	k
Level 6	/	R₀
Level 5	/	V_R
Level 4	(Omega)_rad	t₀
Level 3	(Omega)_lambda	D_L
Level 2	(Omega)_mat	D₀
Level 1	z	D

Example:     With    (Omega)_mat = 0.044 , (Omega)_lambda = 0.519 , (Omega)_rad = 0.000049   &   z = 7

   4 FIX

     0.000049 ENTER
        0.519     ENTER
        0.044     ENTER
            7        ITG D = 14.1191
D₀ = 34.1900
   D_L = 413.9227
   t₀ = 15.5877
V_R = 2.4826
   R₀ = 20.8340
k = -1

Note:

-The precision may be better than with Z->D if z is very small

5°) Carlson Elliptic Integrals RF RJ RFZ RJZ

RF calculates: R_F(x;y;z) = (1/2) §₀^infinity ( ( t + x ).( t + y ).( t + z ) )^-1/2 dt with x , y , z non-negative and at most one is zero

RJ calculates: R_J(x;y;z;p) = (3/2) §₀^infinity ( t + p )^-1 ( ( t + x ).( t + y ).( t + z ) )^-1/2 dt with x , y , z non-negative and at most one is zero and p > 0

RFZ computes R_F ( x , y+i.z , y-i.z )

RJZ computes R_J ( x , y+i.z , y-i.z , p ) with p > 0

STACK	INPUTS	OUTPUTS
Level 3	x	/
Level 2	y	/
Level 1	z	R_F( x , y , z )

Example:

1 ENTER
2 ENTER
4 RF yields R_F( 1 , 2 , 4 ) = 0.685085816632

STACK	INPUTS	OUTPUTS
Level 4	x	/
Level 3	y	/
Level 2	z	/
Level 1	p	R_J( x , y , z , p )

Example:

1 ENTER
2 ENTER
4 ENTER
7 RJ gives R_J( 1 , 2 , 4 , 7 ) = 0.147854444980

STACK	INPUTS	OUTPUTS
Level 3	x	/
Level 2	y	/
Level 1	z	R_F( x,y+i.z,y-i.z )

Example:

1 ENTER
2 ENTER
4 RFZ yields R_F( 1 , 2+4.i , 2-4.i ) = 0.631488738054

STACK	INPUTS	OUTPUTS
Level 4	x	/
Level 3	y	/
Level 2	z	/
Level 1	p	R_J( x,y+i.z,y-i.z,p )

Example:

1 ENTER
2 ENTER
4 ENTER
7 RJZ gives R_J( 1 , 2+4.i , 2-4.i , 7 ) = 0.126039065103

Remark:

DL is a subroutine that is called to calculate the luminosity-distance.

References:

[1] Stamatia Mavridès - "L'Univers relativiste" - Masson ISBN 2-225-36080-7 ( in French )
[2] Jean Heidmann - "Introduction à la cosmologie" - PUF ( in French )
[3] David F. Crawford - "Curvature Cosmology" - ISBN 1-59942-413-4
or http://www.davidcrawford.bigpondhosting.com/cc2.pdf
[4] J. Pachner - "An Oscillating Isotropic Universe without Singularity" - Mon. Not. R. astr. Soc. ( 1965 ) 131, 173-176
[5] Hua-Hui Xiong, Yi-Fu Cai, Taotao Qiu, Yun-Song Piao, Xinmin Zhang - "Oscillating universe with quintom matter"
[6] B.C. Carlson, "Numerical Computation of Real or Complex Elliptic Integrals"
[7] Jacques Colin, Roya Mohayaee, Mohamed Rameez, and Subir Sarkar - "Evidence for anisotropy of cosmic acceleration" - Astronomy & Astrophysic 631, L13 (2019)
[8] Helge Kragh - Cyclic models of the relativistic universe: the early history.
[9] Jean-E. Charon - Complex Relativity ( a controversial unitary theory, but with many interesting ideas )